what I want to do is customize the actual email notification that goes out. For example, I want to add an URL to the email. This does not address that.
If you're customizing the content of your message, I am afraid there is no such way to achieve this at this moment.
We could only set the notifications for the Approve.
As a workaround, we could add a stage at first of the release pipeline, and add the powershell task to sent the e-mail:
Send simple email in Azure DevOps using PowerShell (without any smtp server)
or we could use the send mail task, you could check another thread for some more details:
Best way to send email notification in an Azure DevOps build pipeline? Do not want to use "Notifications Module"